A dual variational principle for the Moser-Trudinger inequality: existence, uniqueness and stability of extremals

For $p>1$, with $p'=p/(p-1)$, we prove the sharp dual inequality $$ \int_0^1-\log\left(1-\frac{\left|\int_0^xf(y) dy\right|^{p'}}{x}\right) dx\leq\mathsf{K}_p, \qquad \|f\|_{L^p(0,1)}\leq1, $$ where $\mathsf{K}_p$ is the logarithm of the optimal value in Moser's one-dimensional formulation of the Mo